Decic

Decic is a temperament for the 7, 11, 13, and 17 prime limits. It is a member of marvel temperaments, cloudy clan, and linus temperaments. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a meantone fifth, but four of them are not used to reach the 5th harmonic. Instead, 14/13, 15/14 and 16/15 are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the magic major third (flat ~5/4) can be used as a generator.

Temperament data

Decic temperament (10&50)

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 16 23 28 35 37 41], ⟨0 1 -1 0 2 0 1]]

POTE generators:

• 7-limit: ~3/2 = 698.69596
• 11-limit: ~3/2 = 696.79119
• 13-limit: ~3/2 = 696.99342
• 17-limit: ~3/2 = 697.08527

TOP generators:

• 7-limit: ~15/14 = 120.18411, ~3/2 = 699.76795
• 11-limit: ~15/14 = 120.14165, ~3/2 = 697.61366
• 13-limit: ~14/13 = 120.11775, ~3/2 = 697.67733
• 17-limit: ~14/13 = 120.12744, ~3/2 = 697.82559

Diamond monotone ranges:

• 7-odd-limit: ~3/2 = [680.00000, 720.00000] (34\60 to 6\10)
• 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
• 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)

Diamond tradeoff ranges:

• 7 and 9-odd-limit: ~3/2 = [693.12909, 702.51219]
• 11, 13, and 15-odd-limit: ~3/2 = [689.36294, 702.51219]
• 17-odd-limit: ~3/2 = [689.36294, 704.95541]

Diamond monotone and tradeoff ranges:

• 7-odd-limit: ~3/2 = [693.12909, 702.51219]
• 9-odd-limit: ~3/2 = [696.00000, 702.51219]
• 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]

Optimal GPV sequences:

• 7-limit: 10, 30b, 40, 50, 60, 110d, 170cdd
• 11 and 13-limit: 10, 30be, 40, 50
• 17-limit: 10, 30beg, 40, 50

Badness:

• 7-limit: 0.089135
• 11-limit: 0.063900
• 13-limit: 0.036880
• 17-limit: 0.025064